Problem 10 - Algebra 1

Question

You work a different number of hours each day. The table shows your total pay p and the number of hours h you worked. $$\begin{array}{|l|c|c|c|c|}\hline \text { Total pay, } p & \$ 18 & \$ 42 & \$ 48 & \$ 30 \\\\\hline \text { Hours worked, } h & 3 & 7 & 8 & 5 \\\\\hline \text { Ratio } & ? & ? & ? & ? \\\\\hline\end{array}$$ Write a model that relates the variables p and h.

Step-by-Step Solution

Answer

The model that relates the pay p to the hours h worked is given by p = rh, where r is the rate of pay per hour. If the ratio calculated in steps 1 and 2 is consistent across all the (p, h) pairs, r is the value of that constant ratio.

Step 1: Calculate each ratio

Calculate the ratio of pay to hours worked for each pair (p, h) by dividing the total pay by the number of hours. This gives: 18/3, 42/7, 48/8, and 30/5.

Step 2: Check if ratios are equal

Check if all ratios obtained in the previous step are equal. If they are, it means the pay is proportional to the hours worked and the ratio is the rate of pay.

Step 3: Formulate the model

If the ratio is constant, then a linear relationship can be established. This relationship can be modeled by the equation p = rh, where r is the constant ratio (rate of pay per hour) and h is the number of hours worked.

Key Concepts

Ratio and Proportion

In algebra, a ratio is a comparison of two quantities that expresses the magnitude of one in relation to the other. It is often written in the form of a fraction, such as \( \frac{a}{b} \) where \( a \) and \( b \) are two entities being compared. When we talk about ratio and proportion, we are looking at the correspondence between two ratios.

Proportion, on the other hand, indicates that two ratios are equivalent. This means that when we scale up one quantity, the other quantity scales in a way that the ratio remains constant. In the context of your homework problem, you are comparing total pay \( p \) to hours worked \( h \) and examining whether this ratio is consistent across different values.

• If the ratio \( \frac{p}{h} \) remains the same for all entries, then \( p \) and \( h \) are in direct proportion.
• Understanding proportions allows us to predict one quantity based on another, given the constant ratio or 'rate'.

When you're finding the ratio for each pair (total pay and hours worked), you're seeking to identify the constant rate of pay per hour, which is central to understanding salaries, wages, and many other real-world applications of ratios.

Modeling with Equations

Modeling with equations is an important aspect of algebra that involves representing real-world situations mathematically. In your exercise, the model you are tasked with deriving is a way to describe the relationship between total pay \( p \) and hours worked \( h \) using an algebraic equation.

To achieve this, you already computed individual ratios to find out if total pay is directly proportional to hours worked. Once you've verified that the ratios are indeed equal (which is your constant rate of pay per hour), you can model the situation with a linear equation.

Form of the Model

In its simplest form, the linear relationship is expressed as \( p = rh \) where \( r \) denotes the constant ratio, the rate of pay per hour. This model efficiently encapsulates the relationship and allows for different operations such as:

• Predicting future pay for a given number of hours worked
• Calculating hours needed to earn a desired amount

By using variables and equations to model these situations, algebra becomes a powerful tool for solving problems and making informed decisions in various contexts.

Variables in Algebra

In algebra, variables are symbols that represent values which can change within the scope of a given problem. In your exercise, \( p \) and \( h \) are variables representing total pay and hours worked, respectively.

Variables allow us to formulate expressions and equations that generalize the relationships between differing quantities. When you formulate an equation like \( p = rh \) after determining a consistent ratio, you're actually defining a function where \( p \) varies with \( h \) in a specific, predictable manner.

Utilizing Variables

• \textbf{Representation:} Variables stand in for unknown or dynamic quantities, enabling us to describe and analyze relationships.
• \textbf{Manipulation:} By performing algebraic operations on variables, we can solve for unknown values, transform equations, and explore the implications of different values within the given context.
• \textbf{Flexibility:} Once an equation is set up with variables, it can be used for a wide range of numbers, making variables a fundamental concept for modeling real-world scenarios algebraically.

Understanding how to work with variables is crucial to mastering algebra and applying it effectively to solve problems, both in academia and in everyday life.